Throughout the construction we shall repeatedly use the well-known fact (see the Defect Theorem in [1]} that for two words \(w_1,w_2\) , we have \(w_1w_2 = w_2 w_1\)  if and only if they are repetitions of the same word, that is there exists a word \(w_3\)  and \(p,q \in \mathbb {N}\)  such that \(w_1 = w_3^p\)  and \(w_2 = w_3^q\) .
